(c) Suppose R90D and R1Y are both non-stationary. Hence, you decide to run an OLS regression
of one variable on the other and obtain the following parameter estimates and adjusted R^(2) :
widehat(R1Y)_(t)=0.361+1.046R90D_(t)(,)/(b)ar (R)^(2)=0.973
Next, you perform an ADF test on the estimated residuals hat(u)_(t) from this regression:
[, test stat ,p-value ],[hat(u)_(t),-6.974,0.006]
Based on a significance level of 5%, what do you conclude from the results? Does the result
suggest to take only first differences of R90D and R1Y ?
[Do not write more than six lines of text.]
(15 points)
(d) A second implication of the expectations theory (which was mentioned at the beginning of
this exercise) is that the spread between the two interest rates will be cointegrated with
cointegrating coefficient gamma =1. Hence, you run an ADF test on the spread between R1Y and
R90D :
Next, you specify a vector error correction model (VECM) for the two interest rates using
an error correction term with the theoretically suggested value of gamma =1. Hence, you add the
lagged spread, R1Y_(t-1)-R90D_(t-1) to a two-dimensional VAR(2). For the VECM, you obtain
the following parameter estimates and t-statistics (given in parentheses):
Based on a significance level of 5%, what do you conclude from results of the ADF test and
the VECM for this second implication of the expectations theory?
[Do not write more than eight lines of text.]
(20 points)
Multiple-choice
points
(c) Suppose R90D and R1Y are both non-stationary. Hence, you decide to run an OLS regression of one variable on the other and obtain the following parameter estimates and adjusted R2: R1Y = 0.361 +1.046 R90D, =0.973.
Multiple-choice
(30 points)
4. There are given five statements per block below. Only two of these statements are true. You will get
Next, you perform an ADF test on the estimated residuals u, from this regression:
6 points if you select the two correct statements. 2 points if you select one correct statement. 0 points if you select only wrong or more than two statements. SELECT ONLY TWO STATEMENTS in each of the following problems. No explanations need to be given.
test stat p-value u 6.974 9000
Based on a significance level of 5%, what do you conclude from the results? Does the result suggest to take only first differences of R90D and R1Y? [Do not write more than sir lines of text./ (15 points) (d) A second implication of the expectations theory (which was mentioned at the beginning of this exercise) is that the spread between the two interest rates will be cointegrated with cointegrating coefficient = 1. Hence, you run an ADF test on the spread between R1Y and R9OD:
Block A: Properties of the OLS estimator
(6 points)
I If the following assumptions hold, the OLS estimators are known as Best Linear Unbiased Estimators (BLUE) E ( =0, Vr (u)=<o, and Cov (u)=0. (II) The OLS estimators and are linear estimators. (III) The OLS estimators are unbiased. This implies that on average the estimated values will be equal to their true value, i.e., E() = and E() = . (IV) The OLS estimators are best in the sense that any alternative unbiased estimator has a smaller variance. (V) The OLS estimators are consistent. This implies that the estimated values will converge to their true values as the sample size increases to infinity.
test stat p-value R1Y - R90D 6.312 0.008
Next, you specify a vector error correction model (VECM) for the two interest rates using an error correction term with the theoretically suggested value of = 1. Hence, you add the lagged spread, R1Y-1 R90D,1 to a two-dimensional VAR(2). For the VECM, you obtain the following parameter estimates and t-statistics (given in parentheses):
R90D=0.14 0.24 R90D-1- 0.44 R90D,2 (0.82)(0.75) (1.29) 0.01 RIY + 0.15 RIY=0.18 [R1Y1 R90D] (0.03) (0.56) (0.67)
Block B: Classical linear regression model (6 points) (I) The R2 is related to the correlation coefficient between z and y and can therefore vary between -1 and 1. (II) If you re-parameterise a regression model, the R2 will remain unchanged. (III) If there is evidence for heteroscedasticity, the parameter estimates and the standard errors might be wrong. (IV) If there is evidence for autocorrelation, the standard errors might be wrong. (V) If there is perfect multicollinearity, you cannot estimate the regression at all.
RIY,= 0.36 0.14 R90D,- 0.33 R90D,2 (2.25)(0.47) (1.14) 0.11 RIY1 +0.10 R1Y=0.52[RIY-=R90D] (0.31) (0.40) (2.17)
Based on a significance level of 5%, what do you conclude from results of the ADF test and the VECM for this second implication of the expectations theory? Do not write more than eight lines of tert./ (20 points)
Block C: MA, AR, and ARMA processes (6points) (I) The autocorrelation function of an MA(q) process is geometrically declining. (II) The partial autocorrelation function of an MA(q) process is geometrically declining. (III) The autocorrelation function of an AR(p) process is zero after lag p. (IV) The partial autocorrelation function of an AR(p) is zero after lag p. (V) If you follow the rolling window approach to forecast a model, you keep the starting point of your estimation period fixed, so that it becomes larger over time.
